Indentation test of soft tissues with
curved substrates: A finite
element study
M. H. Lu
Y . P . Zheng
Rehabilitation Engineering Center, The Hong Kong Polytechnic University, Hong Kong, China
Abstract--Indentation is a c o m m o n l y used approach to measure the mechanical
properties of soft tissues, such as articular cartilage and limb tissues. The Young's
modulus of tissue can be calculated from the indentation test using a mechanical
model, where the soft tissue is normally assumed to have a flat substrate. In this
study, a series of 2D finite element models were established to investigate the effects
of bones with various curvatures embedded in the soft tissues during an indentation
test. For each curvature of the hard substrate, the errors in the calculation of the
Young's modulus were estimated for different indentation depths (0-10%) and aspect
ratios a/ h of the indenter diameter and the tissue thickness (0.2-2 in seven steps). The
radius ratio a/R of the indenter and the curved substrate ranged from 0 to 0.38 in nine
steps. Results showed that the error in calculation of the Young's modulus increased
by 21.2% when the curvature a/R of the bone increased from 0 to 0.38 (under the
condition of a / h - 1.0, Poisson's ratio v - 0.45). The error increased from 6.0 to 18.2%
when the tissue thickness increased from 0.2 to 2 (a/R-0.18, v - 0.45). It was found
that the error in the Young's modulus calculation caused by the curved hard substrates
could be corrected by a single factor for different indentation depths. This factor
depends on the Poisson's ratio, the aspect ratio a/h and the radius ratio a/R.
Keywords--lndentation, Ultrasound indentation, Soft tissue assessment, Finite element
analysis, Limb tissues, Residual limbs
Med. Biol. Eng. Comput., 2004, 42, 535--540
\
J
1 Introduction
INDENTATIONIS a commonly used approach to measure nondestructively the mechanical properties, especially stiffness, of
soft tissues. During recent decades, it has been widely used for
the assessment of articular cartilage, which is a thin tissue layer
covering the joint bone. With the stimulation of this application,
various models for the indentation on a thin layer of tissue with a
flat supporting surface have been developed. A theoretical
solution to the problem of axisymmetric indentation was
reported (HAYES et al., 1972), where the articular cartilage
bonded to the subchondral bone was modelled as a homogeneous, isotropic and linearly elastic layer bonded to a rigid
half-space (Fig. 1). The Young's modulus E of the cartilage can
be calculated by the derived equation
E =
(1 - - V2)
f
2atc(v, a/h) "w
(1)
where P is the indentation force, v is the Poisson's ratio, w is the
indentation depth, a is the radius of the indentor, h is the
Correspondence should be addressed to Dr Yongping Zheng;
emaih rczheng@polyu.edu.hk
Paper received 28 October 2003 and in final form 19 March 2004
MBEC online number: 20043899
© IFMBE: 2004
Medical & Biological Engineering & Computing 2004, Vol. 42
thickness of the test tissue, and ~c is a geometry and materialdependent factor. Values of tc have been solved for different
values of the aspect ratio a/h (ranging from 0.2 to 2) and
Poisson's ratio v (ranging from 0 to 0.5) (HAYES et al., 1972;
JURVELINet al., 1990). These results have been widely used as a
theoretical background in the measurement of the mechanical
properties of articular cartilage. A new mathematical method
was reported to study the indentation problem of a tissue
layer overlaid on a rigid foundation (SAKAMOTO et al., 1996).
Closed-form solutions of the function ~cwere obtained through
an infinite series.
However, in the investigations mentioned above, the indentation depth was assumed to be very small, and the effects
of the curved substrate geometry were neglected. This is not
true for most of the experimental studies. The influence of a
large deformation on the calculation of the Young's modulus
from the rigid, cylindrical, flat-ended indentation test of
soft tissues was studied using finite element analysis (ZHANG
et al., 1997). A new set of ~c values was reported for the
calculation of Young's modulus to account for the effects of
large deformation.
GALBRAITH and BRYANT (1989) studied the effect of
model geometry and deformable substrate of articular surface
using a linear elastic finite element model. They indicated that
the results of the indentation analysis were unaffected if the
radius of the tissue was at least three times the radius of
the indenter, and if the thickness of the deformable substrate
535
force
force
?
2
3L~
softtissue
t
h
1
Fig 3
Fig 1
Schematic diagram o f indentation problem including rigid,
flat-ended cylindrical indentor and soft-tissue layer bonded
with rigid flat substrate, wherea is" radius" o f indento~ h is"
thickness of test tissue, and w is" indentation depth. There was
no friction between indentor and tissue surface
was at least 16 times the tissue thickness. However, they
did not study the effects of curved rigid substrates for the
indentation tests.
In addition to articular cartilage, the Hayes indentation model
has also been widely used for the measurement of the mechanical properties of limb tissue (ZHEN6 et al., 2001), postirradiation neck fibroid tissue (Ltmy6 et al., 2002), plantar
foot tissue and other soft tissues (ZHEN6 et al., 2000;
KLAESNERet al., 2002). The development of ultrasound indentation approaches has made the Hayes model more acceptable, as
the tissue thickness can be obtained non-destructively during the
indentation test and used to calculate correct ~ values (ZHEN6
and MAK, 1996; ZHEN6 and MAK 1999; SUN et al., 2001;
LAASANEN et al., 2002; HAN et al., 2003).
In the ultrasound indentation, an ultrasound transducer is used
as the indentor and, at the sametime, to measure the tissue
thickness and indentation depth. The influences of indentor
misalignment, indentation rate and muscular contraction on
the calculation of Young's modulus have been previously
investigated (ZHENG et al., 1999). BOSBOOM et al. (2001) and
OOMENS et al. (2003) constructed 3D models of indented soft
tissues overlying bony prominences, representing realistic
bone-tissue geometries, to study the mechanical properties of
skeletal muscles. However, the effects on the indentation
responses of those bones embedded in the soft tissue have not
been quantitatively assessed yet.
In this study, we investigated this issue using finite element
analysis so as to understand how large errors could be caused in
calculation of the Young's modulus if the indentation model for
the flat substrate is used for the indentation test for the curved
~h
r
substrate; study these errors and find potential approaches to
correct them.
2 Methods
A series of axisymmetrical 2-D finite element models were
established using ABAQUS (Version 6.2, Hibbitt, Karlsson &
Sorensen, inc, US). The models included a rigid, flat-ended,
cylindrical indentor, a soft tissue layer and a rigid object with
various curvatures mimicking different sizes of bones as shown
in Fig. 2. The soft tissue was assumed to be homogeneous and
isotropic and adhered to the rigid bony foundation. The soft
tissue was assumed to be linearly elastic. A typical Young's
modulus of 60 kPa (ZHENGet al., 2001) was assigned for the soft
tissues, and a Poisson's ratio of 0.45 was used considering the
nearly incompressible feature of soft tissues under a rapid
indentation (ZHENG et al., 2001). The Young's moduli of the
bone and the indentor were assigned to be 17 GPa and 100 GPa,
and Poisson's ratio of 0.3 and 0.4, respectively (FUNs, 1993;
force
q
indentor(a=4.5mm)
h=11.25mm
~ , ~
force
softtissue
Finite element mesh for axisymmetric indentation problem
including rigid flat-ended cylindrical indentor and soft-tissue
layer bonded with rigid flat substrate (a/h = 0.4, a = 4.5 mm,
h = ll.25mm, where a is" radius o f indento~ and h is" thickhess of test tissue). There was" no friction between indentor
and tissue surface
tissue
Jl
t
softtissue
radius R of bone,
Fig 2 Schematic o f indentation problem including rigid flat-ended
cylindrical indentor and soft-tissue layer bonded with rigid
curved substrate, where a is" radius o f indento~ h is" thickness
o f test tissue, and w is" the indentation depth. There was no
friction between indentor and tissue surface
536
Fig 4
mm
Series" o f indentation models" with curved substrate embedded
in tissue. Radius of bone ranged from 54mm to 12mm, i.e.
a/R ranged from 0.083 to 0.375 (a/h=0.4, a = 4 . 5 m m ,
h = ll.25mm, where a is radius ofindento~ andh is" thickness
of test tissue) where R is" radius" o f substrate bone
Medical & Biological Engineering & Computing 2004, Vol. 42
Table l Comparison o f ~: values calculated by present model and those reported by HArEs et al. (1972) and
ZHANG et al. (1997) (where a is radius o f indento~ h is the thickness o f test tissue, and w is indentation depth)
a/h
0.2
0.4
0.6
0.8
w/h, %
1.0
2.0
1.5
~:
0
HAYES et al. (1972)
0.1
ZHANG et al. (1997)
present model
1.0
ZHANG et al. (1997)
present model
5.0
ZHANG et al. (1997)
present model
10.0
ZHANG et al. (1997)
present model
1.252
1.599
2.031
2.532
3.085
4.638
6.380
1.233
1.246
1.625
1.601
2.035
2.038
2.496
2.545
3.064
3.106
4.734
4.693
6.478
6.478
1.240
1.245
1.632
1.606
2.048
2.051
2.520
2.568
3.090
3.137
4.785
4.749
6.550
6.558
1.267
1.225
1.666
1.621
2.108
2.093
2.621
2.644
3.213
3.250
5.026
4.958
6.893
6.874
1.304
1.218
1.706
1.640
2.181
2.146
2.751
2.739
3.362
3.392
5.317
5.219
7.307
7.269
ZHANG et eL, 1997). The large deformation effect was considered and no friction between the indentor and the tissue surface
was assigned.
A concentrated force was applied at the center of the upper
surface o f the indentor to simulate the indentation. The indentation depth under an assigned force was calculated using the
finite element analysis. By a given Poisson's ratio, the Young's
modulus o f the soft tissue was calculated using equation (1). The
geometry and material-dependent factor ~ in equation (1)
changes with the aspect ratio e / h , the Poisson's ratio v and the
relative deformation o f the tissue w / h (HAYES et el., 1972;
ZHANG et el., 1997). To use the proper ~ in the calculation o f the
Young's modulus in equation (1) for large deformation correction, a finite element model with a flat substrate was firstly
established (Fig. 1). According to the tissue ultrasound palpation
system used in measuring the Young's moduli of soft tissues, the
radius o f the indentor was assigned as 4.5 m m (ZHENG et el.,
1999). A typical range o f tissue thickness o f 2.25-22.5 m m was
studied (with the aspect ratio e / h ranging from 0.2 to 2 for each
case). When finite element analysis is used, grid density is an
important consideration in determining the solution accuracy.
According to the selection criteria reported by ZHANG et el.
(1997), a grid o f 0.5 x 0.5 m m 2 was selected for the simulation
(Fig. 3) after comparison of the performances of grids o f
0.3 x 0 . 3 m m 2, 0.5 x 0 . 5 m m 2 and 1.5 x 1.5mm 2. To validate
the finite element model, a series o f calculations for ~ were
carried out for different indentation depths. These data for
were used to calculate the Young's modulus o f the tissue bonded
to the curved foundation.
To study the effects o f different curvatures o f bones
embedded in the limb tissues, the diameter o f the curved rigid
inclusion was selected to change from 54 to 12 m m in eight
steps, i.e. e / R ranged from 0.083 to 0.375 (Fig. 4). Different
mesh solutions were tried by changing the grid density, grid
shape and solution algorithm in ABAQUS. After comparing
different meshing approaches for the calculation time, partition
complexity and maximum deformation that could be modelled,
we selected the mesh with a 0.5 x 0.5 m m 2 grid, a quadraticstructured shape for the indentor and the upper 20% o f the tissue
and a quadratic, free-advancing front shape for the lower 80% o f
the tissue and bone. All the meshed models were checked and
verified by the functions provided by ABAQUS.
For the simtflation process, different increment steps were
tried to find whether more steps could increase the accuracy o f
the result analysis. For the indentation with a small deformation,
a small increment was used for the simulation (for w / h = 1%, ten
Medical & Biological Engineering & Computing 2004, Vol. 42
increment steps), whereas, for a large deformation, a relatively
larger increment was assigned (for w / h = 10%, 50 increment
steps). For each curvature of the bone, a series o f finite element
models were established for the aspect ratio e / h ranging from
0.2 to 2. For each simulation, the indentation was achieved by
assigning the applied force. The Young's modulus of the tissue
was calculated using the indentation depth and ~ obtained from
the reference model with a flat substrate.
3 Results
Table 1 gives a comparison between the ~ values calculated
by the model o f the present study and those reported by ZHANG
et el. (1997) and HAYES et el. (1972). it was found that, when a
small deformation (i.e. w / h = 0.1%) was applied, the ~ values
obtained in this study agreed very well with the data reported by
HAYES et el. (1972) and ZHANG et el. (1997). When a large
deformation (e.g. w / h = 10%) was applied, the ~ values showed
a good agreement with the data reported by ZHANG et el. (1997).
The maximum difference was within 6%.
58.057.557.056.556.0~_ 55.555.054.5
&.A,.,---A
A
~
A----____._A
~
~
1'0
1'2
deformation, %
Fig 5
Estimated Young's modulus o f tissue (a/h = 0.6, a/R = 0.18,
where a is radius o f indento~ h is thickness o f test tissue, and
R is radius ofsubstrate bone). Assigned Young's modulus was
60 kPa. (- x -) indicated the results calculated using ~: values
given by HAYES et al. (1972), with large deJbrmation effect
included," (-&-) indicated the values calculated using tc
values obtained for identation with flat substrate, with large
deformation effect corrected
537
60-
a/R= 0
Fig. 5 shows the difference in the derived Young's modulus
for a typical model (a/h = 0.6, a / R = 0.18), before and after the
~c values for the large deformation effect were corrected. When
those ~cvalues obtained from the simulation of the indentation on
the tissue with the flat substrate were used for this model, the
calculated Y o u n g ' s moduli of the tissue kept a nearly constant
value, indicating that the effect of large deformation was
corrected. However, the calculated Young's moduli decreased
when the curvature of the bone increased.
For the same aspect ratio a/h of 0.6, the effect of the
embedded bone shape was studied, and a typical set of data is
shown in Fig. 6. The error of the derived Young's modulus
increased significantlywhen the curvature of the bone increased.
Such an effect was compared for different aspect ratios a/h and
is shown in Fig. 7. it was obvious that the calculation error of the
modulus caused by the curved substrate also increased when the
aspect ratio increased. Different amounts of error in the extracted
modulus could be induced by the curvature of the bone for the
thin layer of tissue. For example, when a/h = 2.0, the derived
Y o u n g ' s modulus of the test tissue with an embedded bone
( a / R = 0 . 3 8 , i.e. R = 12ram) was 40.8kPa, which was 32%
smaller in comparison with the assigned Young's modulus of
60 kPa. The results suggested that a modified set of ~c values
should be used in calculating the Young's modulus, using (1)
to correct the error caused by the curvature of the rigid inclusion.
We proposed another factor 7(v, a/h, a/R) to correct this
error (equation (2)). The values of 7 for v = 0.45 are given in
Table 2.
58-o
o--
o
56- ,-~, ~ - -
o
[]
~
u
~
-~¢ ~ ~
E
54-
52-
-:
: --
-o
o--o
- ,
, --
'~
~
o ~ . ~
aiR=0.08
[] ~ . - D
~ ~
aiR=0.11
a/R=0.13
aiR=0.15
~ ~
a/R=0.18
~
~
~
~
o
'
,
~
a/R=0.23
-~-
a/R=0.30
a / R = 0.38
5O
;
o
~
&
~
;
}
;
;o
;
1'1
relative deformation w/h, %
Fig 6
Effect of curved substrate embedded in soft tissue (a/h = 0.6,
where a is" radius" of indento~ and h is" thickness of test tissue).
Captions represent ratios of indentor radius and bone curvature. Young's moduli were calculated using ~cvalues obtained
with consideration of large deJbrmation effect
. 35-
~
~
a/h=2.0
30E
~o
-c~ 2 5 -
a / h = 1.5
a / h = 1.0
~
20-
a/h=0.8
a/h=O.6
15-
a / h = 0.4
a / h = 0.2
c
lOg
{3.
(1 - v 2)
P
E = 2atc(v, a/h, w / h ) . 7(v, a/h, a/R) "w
50
0.05
0.10
0.15
0.20
0.25
0.30
0.35
The error reduction in Young's modulus estimation after
application of(2) can be calculated using the following equation
(suppose Eo is the Young's modulus before 7 correction, and E 1
is the Y o u n g ' s modulus after 7 correction, then E 1 ----Eo/7):
0.40
a / R (indentor radius/bone radius)
Fig 7
(2)
A - IE°
- -- Ell x 100% = IE°(1 - (1/7))1 x 100%
Comparison of errors" of derived Young's moduli of tissues
embedded with substrates with diffbrent curvatures" when
aspect ratio ranged from 0.2 to 2. Error bars" represent
standard deviation of resuhs of seven calculation points"
with diffbrent indentation depths (J?om 0.1 to 10%)
Eo
=
Eo
1 - - 1 ×100%
7
(3)
Table 2 7 values used to correct error in calculation of Young's modulus caused by curved hard substrate in
indentation problem. Poisson's ratio of tissue was assigned as 0. 45 (where a is radius of indento~ h is thickness of
test tissue, and R is radius of substrate bone)
a/R=O
0.08
0.11
0.13
0.15
0.18
0.23
0.30
0.38
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.97
0.96
0.96
0.95
0.94
0.92
0.91
0.96
0.95
0.94
0.93
0.92
0.90
0.88
0.96
0.95
0.94
0.92
0.91
0.89
0.86
0.95
0.94
0.93
0.91
0.90
0.87
0.84
0.94
0.93
0.91
0.90
0.88
0.85
0.82
0.93
0.91
0.89
0.88
0.86
0.82
0.78
0.90
0.89
0.87
0.84
0.82
0.77
0.73
0.88
0.87
0.84
0.81
0.79
0.73
0.68
a/h = 0.2
0.4
0.6
0.8
1.0
1.5
2.0
Table 3 Pereentage error reduction in Young's modulus after 7 correction is applied (where a is radius of indento~ h is
thickness of test tissue, and R is radius of substrate bone)
a/h = 0.2
0.4
0.6
0.8
1.0
1.5
2.0
538
aiR = 0
0.08
0.11
0.13
0.15
0.18
0.23
0.30
0.38
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3.06
3.76
4.44
5.41
6.16
8.12
10.07
3.94
5.14
6.07
7.28
8.26
11.10
13.77
4.71
5.67
6.89
8.26
9.33
12.71
15.68
5.42
6.72
7.99
9.50
11.09
14.81
18.36
6.39
7.97
9.51
11.35
13.27
17.79
22.23
8.03
9.74
11.74
14.12
16.44
22.19
27.87
10.99
12.67
15.39
18.65
21.61
29.68
37.41
13.96
15.60
18.92
22.85
26.88
36.99
47.02
Medical & Biological Engineering & Computing 2004, Vol. 42
Table 3 summarises the error reduction in Young's modulus
after 7 correction is applied.
4 Discussion
in this study, we investigated the effects of the curved
substrate on the indentation response of a soft-tissue layer
using finite element analyses. A series of 2D finite element
models were established to mimic bones of different sizes
embedded in the soft tissues. When a traditional indentation
model derived for a flat substrate (HAYESe t al., 1972) was used
to calculate the Young's modulus from the indentation response
of these models, errors could be induced. These errors were
found to be independent of the indentation depths after the large
deformation efforts were corrected (ZHANGe t al., 1997). Results
also showed that the error in calculating the Young's modulus
increased when the curvature of the bone increased and when the
tissue thickness decreased for a certain curvature.
A modified version of the Hayes indentation equation was
proposed to include a factor to correct the error caused by the
curved substrate. This factor depends on the Poisson's ratio v of
the tissue, the radius ratio a/R of the indentor and the curved
substrate, and the aspect ratio a/h of the indentor radius and
the tissue thickness. Corrections for the errors induced by the
curved substrate can be made using the proposed method,
together with the estimation of the bone curvature using some
topography-imaging methods, such as CT, B-mode ultrasound,
MRI etc. Measurement of the tissue thickness can be achieved
using A-mode ultrasound (ZHENGe t al., 1996; HAN et al., 2003;
SUH et al., 2001; LAASANENet al., 2003).
In this study, a Poisson's ratio of 0.45 was assigned for the
soft tissue, assuming it is nearly incompressible under a rapid
indentation (HAYES et al., 1972; ZHENG et al., 2001; LAASANEN
et al., 2002). To obtain the Poisson's ratio of the tissue in vivo has
been an engineering challenge for many years. Most recently,
attempts have been made to measure the Poisson's ratio of tissue
non-destructively, using ultrasound elastography (RIGHETTI
et al., 2004). if this technique can be successfully used in vivo,
ultrasound imaging will be able to measure both the bone
curvature and the Poisson's ratio of the soft tissue.
When the tested materials have other values of Poisson's
ratio, the correction factors for the effects of the curved substrate
have to be re-calculated. A more comprehensive set of factors
is being established and can be used in other areas using indentation techniques. We plan to construct a comprehensive set of
~c values for the indentation problem with curved substrate, by
including the effects of the Poisson's ratio v, the aspect ratio
a/h, the radius radio a/R and the relative deformation w/h.
The 2D finite element model used in this study could be
improved so that it better simulated the limb tissues embedded
with bones, as the bones are normally not axisymmetric in the
indentation area. in addition, the curved surface of the limb
tissue could also be modelled in the future. As suggested by
the results of this study, using a small indentor may reduce the
effects of the curved surfaces, as long as it does not cause pain
to the subjects (ZHENGe t al., 1999). Furthermore, limb tissues
are complex in their anatomical structures and can contain
biological tissues with different biomechanical characteristics,
such as tendon, muscle and skin. The two-dimensional homogeneous models used in this study are not enough to describe
these complex structures and mechanical properties.
investigations into modelling realistic bone geometry using
3D models have been reported in the literature (VANNAHand
CHILDRESS,1996; BOSBOOMet al., 2001 ; OOMENSet al., 2003).
However, in most of the experimental indentation studies of limb
soft tissues, the tissues were generally simplified to be homogeneous and isotropic, and the overall mechanical properties
Medical & Biological Engineering & Computing 2004, Vol. 42
were studied. The material properties extracted were quantified
values representing overall tissue stiffness.
In spite of these assumptions, the indentation test is an
improved approach for tissue assessment in comparison with
manual palpation, which can only give a subjective qualitative
judgment for the tissue stiffness. As a first step, we investigated
the effects of the bony curvatures by assuming that the soft
tissues were homogeneous and isotropic. The limitation of such
simplification is that we cannot predict the effect of the
inhomogeneity and anisotropy of the tissues on the loadindentation relationship, if we want to take the tissue inhomogeneity and anisotropy into account, 2D indentation models are
not enough and more realistic 3D models are required.
This study has not addressed the non-linear and visco-elastic
properties of soft tissues. Hence, the effect of non-linear and
visco-elastic properties on the calculated results is uncertain.
in spite of the fact that soft tissues are non-linear and viscoelastic, in most of the experimental indentation studies reported
in the literature, linear elastic models have been commonly used
to simplify the problem, so as to obtain a single 'effective'
modulus of the soft tissues. This single material parameter
represents the average tissue stiffness under different deformations. Hence, as a first step, we assumed that the soft tissues were
linear elastic in our models. The limitation of such a simplification is that we cannot predict the effect of the non-linearity of
the tissues on the load-indentation relationship. Future studies
to address the effects of non-linear visco-elasticity, together with
those of inhomogeneity and anisotropy, should follow.
Another possible error source in our simulation is the 'offaxis' position of the indentor relative to the curved bony
substrate, in such conditions, the curved substrates might act
more as an inclined plane. Furthermore, if the bone is severely
off-centre relative to the indentor, there can be bulk tissue
displacement rather than tissue deformation. Subsequently,
severe errors can be introduced in to the estimation of indentation deformation w. As a further step, we plan to use real surface
and internal geometries of limb tissues, obtained using CT, MRI
or 3D ultrasound approaches, to establish 3D finite element
indentation models to study the effects of the curved surface and
substrate, together with the effects when the indentor is 'off-axis'
relative to the curved substrate.
A number of authors have reported alternative ways to extract
tissue material properties from indentation responses directly
using finite element analyses. REYNOLDS (1988) estimated the
Young's modulus by matching experimental load-indentation
curves with the predictions obtained using finite element
modelling of an indentation into an assumed infinite tissue
layer with idealised material properties. However, the effect of
curved substrates was not investigated in that study.
Other investigators used finite element models of limbs
embedded with bones (originally designed for studying the
interaction between the socket and the residual limb) to
generate reference load-indentation responses (STEEGE,1987;
SILVER-THORN, 1991; 2003; VANNAH and CHILDRESS,1996)
The effects of a curved substrate on the indentation responses
have been inherently included in the calculation. In the case of
complicated geometry and boundary conditions, the direct finite
element method may be a good way to extract the tissue properties
from indentation responses. The relatively long time required
to establish the finite element model for the individual testing
object could be a drawback of such computational methods.
Acknowledgments" This project was partially supported by the
Research Grant Council of Hong Kong (PolyU5199/02E and
PolyU 5245/03E) and The Hong Kong Polytechnic University.
The authors would like to thank Dr Ming Zhang, Jason
Cheung and Xiaohong Jia for their help with the ABAQUS
software.
539
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Authors" biographies
MINHUA LU received her BSc in electronic engineering and information science in 2001 from the University of Science and Technology of
China (USTC). She is currently a PhD student of the Rehabilitation
Engineering Center (REC) in the Hong Kong Polytechnic University.
Her research interests axe in ultrasound measurement and imaging of
tissue mechanical properties.
YONGPING ZHENG obtained his BSc in electronics and information
engineering and his MSc in ultrasound instrumentation from the
University of Science and Technology of China (USTC). He completed his PhD in biomedical engineering at the Hong Kong Polytechnic University in 1997. After spending one year of his
postdoctoral fellowship in acoustic microscope and nonlinear acoustics at the University of Windor, Canada, Dr. Zheng joined the
Rehabilitation Engineering Center (REC), initially as a postdoctoral
fellow and then, in 2001, as am Assistant Professor. His current
research interests include ultrasound elastic measurement and imaging, ultrasound biomicroscopy, elastomicroscopy, 3D ultrasound
imaging of musculoskeletal tissues, sonomyography, and ultrasonic
assessment of articular cartilage.
Medical & Biological Engineering & Computing 2004, Vol. 42